The relationship between two methods for estimating uncertainties in data assimilation

This article examines the relationship between two apparently unrelated methods for estimating error statistics or uncertainties of relevance to data assimilation (DA). The first method is due to Desroziers et al. (2005, Q. J. R. Meteorol. Soc., 131, 3385–3396; referred to as DBCP hereafter) and relies on residual statistics readily available from DA applications. The second method, the three-cornered hat (3CH) developed by Gray and Allan (1974, IEEE 28th Annual Symp. Freq. Control, 243–246) and only recently applied to atmospheric sciences, uses three data sets and can derive estimates of relevant error uncertainties as well. The usefulness of both methods lies in them not requiring knowledge of the true value of the quantities at play. DBCP derives its results by relying explicitly on the constraints associated with the DA minimization problem; 3CH is general and its estimates hold as long as errors in the three data sets of choice are uncorrelated. Establishing the relationship between the methods requires application of the 3CH approach to the same observation, background, and analysis data sets used by DBCP. In this case, the same assumptions of DBCP regarding residual errors allow for cancellation of error cross-covariance terms in 3CH, such that two of its corners derive identical estimates for observation- and background-error covariances to those of DBCP. The error cross-covariance terms associated with the third corner are shown to add up to twice the analysis-error covariance, so that the 3CH estimate for the third corner recovers the negative of the analysis-error covariance. Illustrations of these findings are provided by deriving uncertainties for radio occultation bending angles.